operatingCharacteristics1 {binGroup2}

Calculate operating characteristics for group testing algorithms that use a single-disease assay

Description

Calculate operating characteristics, such as the expected number of tests, for a specified testing configuration using non-informative and informative hierarchical and array-based group testing algorithms. Single-disease assays are used at each stage of the algorithms.

Usage

operatingCharacteristics1(
  algorithm,
  p = NULL,
  probabilities = NULL,
  Se = 0.99,
  Sp = 0.99,
  hier.config = NULL,
  rowcol.sz = NULL,
  alpha = 2,
  a = NULL,
  print.time = TRUE,
  ...
)

opChar1(
  algorithm,
  p = NULL,
  probabilities = NULL,
  Se = 0.99,
  Sp = 0.99,
  hier.config = NULL,
  rowcol.sz = NULL,
  alpha = 2,
  a = NULL,
  print.time = TRUE,
  ...
)

Arguments

algorithm

character string defining the group testing algorithm to be used. Non-informative testing options include two-stage hierarchical ("D2"), three-stage hierarchical ("D3"), four-stage hierarchical ("D4"), square array testing without master pooling ("A2"), and square array testing with master pooling ("A2M"). Informative testing options include two-stage hierarchical ("ID2"), three-stage hierarchical ("ID3"), four-stage hierarchical ("ID4"), and square array testing without master pooling ("IA2").

p

overall probability of disease that will be used to generate a vector/matrix of individual probabilities. For non-informative algorithms, a homogeneous set of probabilities will be used. For informative algorithms, the expectOrderBeta function will be used to generate a heterogeneous set of probabilities. Further details are given under 'Details'. Either p or probabilities should be specified, but not both.

probabilities

a vector of individual probabilities, which is homogeneous for non-informative testing algorithms and heterogeneous for informative testing algorithms. Either p or probabilities should be specified, but not both.

Se

a vector of sensitivity values, where one value is given for each stage of testing (in order). If a single value is provided, sensitivity values are assumed to be equal to this value for all stages of testing. Further details are given under 'Details'.

Sp

a vector of specificity values, where one value is given for each stage of testing (in order). If a single value is provided, specificity values are assumed to be equal to this value for all stages of testing. Further details are given under 'Details'.

hier.config

a matrix specifying the configuration for a hierarchical testing algorithm. The rows correspond to the stages of testing, the columns correspond to each individual to be tested, and the cell values specify the group number of each individual at each stage. Further details are given under 'Details'. For array testing algorithms, this argument will be ignored.

rowcol.sz

the row/column size for array testing algorithms. For hierarchical testing algorithms, this argument will be ignored.

alpha

a shape parameter for the beta distribution that specifies the degree of heterogeneity for the generated probability vector (for informative testing only).

a

a vector containing indices indicating which individuals to calculate individual accuracy measures for. If NULL, individual accuracy measures will be displayed for all individuals in the algorithm.

print.time

a logical value indicating whether the length of time for calculations should be printed. The default is TRUE.

...

arguments to be passed to the expectOrderBeta function, which generates a vector of probabilities for informative testing algorithms. Further details are given under 'Details'.

Details

This function computes the operating characteristics for group testing algorithms with an assay that tests for one disease, as described in Hitt et al. (2019).

Available algorithms include two-, three-, and four-stage hierarchical testing and array testing with and without master pooling. Both non-informative and informative group testing settings are allowed for each algorithm, except informative array testing with master pooling is unavailable because this method has not appeared in the group testing literature. Operating characteristics calculated are expected number of tests, pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value for each individual.

For informative algorithms where the p argument is specified, the expected value of order statistics from a beta distribution are found. These values are used to represent disease risk probabilities for each individual to be tested. The beta distribution has two parameters: a mean parameter p (overall disease prevalence) and a shape parameter alpha (heterogeneity level). Depending on the specified p, alpha, and overall group size, simulation may be necessary to generate the vector of individual probabilities. This is done using expectOrderBeta and requires the user to set a seed to reproduce results.

The sensitivity/specificity values are allowed to vary across stages of testing. For hierarchical testing, a different sensitivity/specificity value may be used for each stage of testing. For array testing, a different sensitivity/specificity value may be used for master pool testing (if included), row/column testing, and individual testing. The values must be specified in order of the testing performed. For example, values are specified as (stage 1, stage 2, stage 3) for three-stage hierarchical testing or (master pool testing, row/column testing, individual testing) for array testing with master pooling. A single sensitivity/specificity value may be specified instead. In this situation, sensitivity/specificity values for all stages are assumed to be equal.

The matrix specified by hier.config defines the hierarchical group testing algorithm for I individuals. The rows of the matrix correspond to the stages s=1,...,S in the testing algorithm, and the columns correspond to individuals i=1,...I. The cell values within the matrix represent the group number of individual i at stage s. For three-stage, four-stage, and non-informative two-stage hierarchical testing, the first row of the matrix consists of all ones. This indicates that all individuals in the algorithm are tested together in a single group in the first stage of testing. For informative two-stage hierarchical testing, the initial group (block) is not tested. Thus, the first row of the matrix consists of the group numbers for each individual in the first stage of testing. For all hierarchical algorithms, the final row of the matrix denotes individual testing. Individuals who are not tested in a particular stage are represented by "NA" (e.g., an individual tested in a group of size 1 in the second stage of testing would not be tested again in a third stage of testing). It is important to note that this matrix represents the testing that could be performed if each group tests positively at each stage prior to the last. For more details on this matrix (called a group membership matrix), see Bilder et al. (2019).

For array testing without master pooling, the rowcol.sz specified represents the row/column size for initial (stage 1) testing. For array testing with master pooling, the rowcol.sz specified represents the row/column size for stage 2 testing. This is because the master pool size is the overall array size, given by the square of the row/column size.

The displayed overall pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value are weighted averages of the corresponding individual accuracy measures for all individuals within the initial group (or block) for a hierarchical algorithm, or within the entire array for an array-based algorithm. Expressions for these averages are provided in the Supplementary Material for Hitt et al. (2019). These expressions are based on accuracy definitions given by Altman and Bland (1994a, 1994b).

The operatingCharacteristics1 function accepts additional arguments, namely num.sim, to be passed to the expectOrderBeta function, which generates a vector of probabilities for informative group testing algorithms. The num.sim argument specifies the number of simulations from the beta distribution when simulation is used. By default, 10,000 simulations are used.

Value

A list containing:

algorithm

the group testing algorithm used for calculations.

prob

the probability of disease or the vector of individual probabilities, as specified by the user.

alpha

level of heterogeneity for the generated probability vector (for informative testing only).

Se

the vector of sensitivity values for each stage of testing.

Sp

the vector of specificity values for each stage of testing.

Config

a list specifying elements of the specified testing configuration, which may include:

Stage1

group size for the first stage of hierarchical testing, if applicable.

Stage2

group sizes for the second stage of hierarchical testing, if applicable.

Stage3

group sizes for the third stage of hierarchical testing, if applicable.

Block.sz

the block size/initial group size for informative Dorfman testing, which is not tested.

pool.szs

group sizes for the first stage of testing for informative Dorfman testing.

Array.dim

the row/column size for array testing.

Array.sz

the overall array size for array testing (the square of the row/column size).

p.vec

the sorted vector of individual probabilities, if applicable.

p.mat

the sorted matrix of individual probabilities in gradient arrangement, if applicable. Further details are given under 'Details'.

ET

the expected testing expenditure to decode all individuals in the algorithm; this includes all individuals in all groups for hierarchical algorithms or in the entire array for array testing.

value

the value of the expected number of tests per individual.

Accuracy

a list containing:

Individual

a matrix of accuracy measures for each individual specified in a. The rows correspond to each unique set of accuracy measures in the algorithm. Individuals with the same set of accuracy measures are displayed together in a single row of the matrix. The columns correspond to the pooling sensitivity, pooling specificity, pooling positive predictive value, pooling negative predictive value, and the indices for the individuals in each row of the matrix.

Overall

a matrix of overall accuracy measures for the algorithm. The columns correspond to the pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value for the overall algorithm. Further details are given under 'Details'.

Note

This function returns the pooling positive and negative predictive values for all individuals even though these measures are diagnostic specific; e.g., the pooling positive predictive value should only be considered for those individuals who have tested positive.

Additionally, only stage dependent sensitivity and specificity values are allowed within the program (no group within stage dependent values are allowed). See Bilder et al. (2019) for additional information.

Author(s)

Brianna D. Hitt

References

Altman, D., Bland, J. (1994). “Diagnostic tests 1: Sensitivity and specificity.” BMJ, 308, 1552.

Altman, D., Bland, J. (1994). “Diagnostic tests 2: Predictive values.” BMJ, 309, 102.

Bilder, C., Tebbs, J., McMahan, C. (2019). “Informative group testing for multiplex assays.” Biometrics, 75, 278–288.

Hitt, B., Bilder, C., Tebbs, J., McMahan, C. (2019). “The objective function controversy for group testing: Much ado about nothing?” Statistics in Medicine, 38, 4912–4923.

McMahan, C., Tebbs, J., Bilder, C. (2012a). “Informative Dorfman Screening.” Biometrics, 68, 287–296.

McMahan, C., Tebbs, J., Bilder, C. (2012b). “Two-Dimensional Informative Array Testing.” Biometrics, 68, 793–804.

See Also

Other operating characteristic functions: GroupMembershipMatrix(), Sterrett(), TOD(), halving(), operatingCharacteristics2()

Examples

# Calculate the operating characteristics for non-informative
#   two-stage hierarchical (Dorfman) testing.
config.mat <- matrix(data = c(rep(1, 10), 1:10),
                     nrow = 2, ncol = 10, byrow = TRUE)
opChar1(algorithm = "D2", p = 0.05, Se = 0.99, Sp = 0.99,
        hier.config = config.mat, print.time = FALSE)

# Calculate the operating characteristics for informative
#   two-stage hierarchical (Dorfman) testing.
# A vector of individual probabilities is generated using
#   the expected value of order statistics from a beta
#   distribution with p = 0.01 and a heterogeneity level
#   of alpha = 0.5.
config.mat <- matrix(data = c(rep(1:3, each = 10), 1:30),
                     nrow = 2, ncol = 30, byrow = TRUE)
set.seed(52613)
opChar1(algorithm = "ID2", p = 0.01, Se = 0.95, Sp = 0.95,
        hier.config = config.mat, alpha = 0.5, num.sim = 10000)
# Equivalent code using a heterogeneous vector of
#   probabilities
set.seed(52613)
probs <- expectOrderBeta(p = 0.01, alpha = 0.5, size = 30)
opChar1(algorithm = "ID2", probabilities = probs,
        Se = 0.95, Sp = 0.95, hier.config = config.mat)

# Calculate the operating characteristics for
#   non-informative three-stage hierarchical testing.
config.mat <- matrix(data = c(rep(1, 18), rep(1:3, each = 5),
                              rep(4, 3), 1:18),
                    nrow = 3, ncol = 18, byrow = TRUE)
opChar1(algorithm = "D3", p = 0.001, Se = 0.95, Sp = 0.95,
        hier.config = config.mat)
opChar1(algorithm = "D3", p = 0.001, Se = c(0.95, 0.95, 0.99),
        Sp = c(0.96, 0.96, 0.98), hier.config = config.mat)

# Calculate the operating characteristics for
#   informative three-stage hierarchical testing,
#   given a heterogeneous vector of probabilities.
config.mat <- matrix(data = c(rep(1, 6), rep(1:2, each = 3),
                              1:6), nrow = 3, ncol = 6,
                     byrow = TRUE)
set.seed(52613)
opChar1(algorithm = "ID3",
         probabilities = c(0.012, 0.014, 0.011, 0.012, 0.010, 0.015),
         Se = 0.99, Sp = 0.99, hier.config = config.mat,
         alpha = 0.5, num.sim = 5000)

# Calculate the operating characteristics for
#   non-informative four-stage hierarchical testing.
config.mat <- matrix(data = c(rep(1, 12), rep(1, 8),
                              rep(2, 2), 3, 4, rep(1, 5),
                              rep(2, 3), 3, 4, rep(NA, 2),
                              1:8, rep(NA, 4)), nrow = 4,
                     ncol = 12, byrow = TRUE)
opChar1(algorithm = "D4", p = 0.041, Se = 0.99, Sp = 0.90,
        hier.config = config.mat)

# Calculate the operating characteristics for
#   informative four-stage hierarchical testing.
# A vector of individual probabilities is generated using
#   the expected value of order statistics from a beta
#   distribution with p = 0.041 and a heterogeneity level
#   of alpha = 0.5.
config.mat <- matrix(data = c(rep(1, 12), rep(1, 8),
                              rep(2, 2), 3, 4, rep(1, 5),
                              rep(2, 3), 3, 4, rep(NA, 2),
                              1:8, rep(NA, 4)), nrow = 4,
                     ncol = 12, byrow = TRUE)
set.seed(5678)
opChar1(algorithm = "ID4", p = 0.041, Se = 0.99, Sp = 0.90,
        hier.config = config.mat, alpha = 0.5)

# Calculate the operating characteristics for
#   non-informative array testing without master pooling.
opChar1(algorithm = "A2", p = 0.005, Se = c(0.95, 0.99),
        Sp = c(0.95, 0.99), rowcol.sz = 8, a = 1)

# Calculate the operating characteristics for
#   informative array testing without master pooling.
# A vector of individual probabilities is generated using
#   the expected value of order statistics from a beta
#   distribution with p = 0.03 and a heterogeneity level
#   of alpha = 2.
set.seed(1002)
opChar1(algorithm = "IA2", p = 0.03, Se = 0.95, Sp = 0.95,
         rowcol.sz = 8, alpha = 2, a = 1:10)

# Calculate the operating characteristics for
#   non-informative array testing with master pooling.
opChar1(algorithm = "A2M", p = 0.02, Se = c(0.95,0.95,0.99),
        Sp = c(0.98,0.98,0.99), rowcol.sz = 5)